Chapter 1 Think as a Macroeconomist: Elements of Micro Behaviour and Macro Modelling ? Jin Cao a,b,* a Munich Graduate School of Economics (MGSE), b Department of Economics, University of Munich, D-80539 Munich, Germany Overview Economic agents in macro models, mainly households and firms, and their decision making problems. Particularly • Modelling decision making problems in continuous time; • Formalizing intertemporal resource constraints; • Properties of neoclassical production functions. Solow-Swan growth model and related issues. Particularly • Dynamics and path of convergence; • Golden rule and dynamic (in-)efficiency; • Stability; • Speed of convergence; • Comparative statics. Mathematical appendix on fundamental calculus: Ordinary differential equations and some useful theorems. ? First version: October, 2007. This version: October, 2008. * Seminar f ¨ ur Makro¨ okonomie, Ludwig-Maximilians-Universit¨ at M ¨ unchen, Ludwigstrasse 28/015 (Rgb.), D-80539 Munich, Germany. Tel.: +49 89 2180 2136; fax: +49 89 2180 13521. Email address: [email protected](Jin Cao). Working Draft 30 December 2008
Transcript
Chapter 1 Think as a Macroeconomist: Elements ofMicro Behaviour and Macro Modelling?
Jin Cao a,b,∗aMunich Graduate School of Economics (MGSE),
bDepartment of Economics, University of Munich, D-80539 Munich, Germany
Overview
Economic agents in macro models, mainly households and firms, and their decision makingproblems. Particularly
• Modelling decision making problems in continuous time;
• Formalizing intertemporal resource constraints;
• Properties of neoclassical production functions.
Solow-Swan growth model and related issues. Particularly
• Dynamics and path of convergence;
• Golden rule and dynamic (in-)efficiency;
• Stability;
• Speed of convergence;
• Comparative statics.
Mathematical appendix on fundamental calculus: Ordinary differential equations and someuseful theorems.
My nearly two decades as a researcher in a central bank taught me two lessons. First,formal analysis, both theoretical and applied, can and does make an immense contributionto improving the conduct of policy. Second, fundamental advances over the last 20 years –say, in basic econometric techniques, in modelling dynamic systems in which expectationsare important, and in data sources – provide an opportunity for historically rapid advancein our understanding of macroeconomic policymaking over the coming years.
— Jon Faust
1 Introduction
This is a warming-up session for our first lecture of Advanced Macro and Money,helping the readers get used to the basic modelling ideas in macro.
Modern macroeconomics is strongly featured by its sound foundation of the microbehaviours. Its methodology is based on the dynamic general equilibrium models,with the potential to accommodate stochastic environments. Therefore the centralissue in this chapter is to explain how to model the micro behaviours of the economicagents, the concepts of which are already familar for the readers, in the macro (dy-namic) set-ups. In the end we apply these settings in a simple partial equilibriummodel, and attempt to obtain some flavour of dynamic equilibrium modellings.
2 Economic Agents and Decision Making Problems
Mostly concerned agents in macroeconomic models are households and firms. Theformer offer labor to earn wage income, and make decisions in consumption andaccumulation of wealth. The latter produce consumption goods by employing laborand investing in capital stock. Therefore to study the behaviour of agents in the econ-omy provides important insights in understanding macroeconomic phenomenen.
2
2.1 Households
2.1.1 Preferences
As you have already learned in intermediate microeconomic theory, the rationalpreference of an economic agent can be captured by a utility function
u(x) : Rn → R
in which x = (x1, . . . , xn) is the bundle containing n goods that gives utility to theagent. For simplicity we assume that the utility function is well-behaved with thefollowing properties (which are actually more than sufficient for the problems inmacroeconomics)
∂u∂xi
> 0,∂2u∂x2
i
< 0,∀i ∈ {1, . . . , n},
i.e. u is strictly concave in all xi’s.
Now in a dynamic context we have to study the agent’s utility function over a periodof time. The simplest way to do that is the approach that you learned in intermediatemacroeconomic theory. Suppose the time is devided into n + 1 discrete periods, andin each interim period t ∈ {0, 1, . . . , n} the agent gains her utility from the bundle xt inthis period. Then the utility function over the entire time span can be expressed as asummation
U =
n∑
t=0
βtu(xt)
=
n∑
t=0
(1
1 + ρ
)t
u(xt)
in which β or 11+ρ act as discount factors. This approach implies that the utility function
is additive with
∂2U∂xit∂xit′
= 0,∀t , t′.
Although such discrete-time approach seems straight-forward for understanding, inpractice the other treatment, continuous-time approach, sometimes provides bettercomputational tractability. Now if we split T into more periods, i.e. we increase
3
n towards infinity, then each period is so small that the time becomes continuous.Therefore when n is infinite the summation above becomes an integration
U =
T∫
0
e−ρtu(x(t))dt
giving a utility function in continuous time. Since readers are less likely to have suffi-cient experience with the continuous-time models, the following sections are mostlywritten in a continuous-time manner. The results from the discrete-time approachare summerized in the end of this chapter, and the developments of these results areleft as your exercises.
2.1.2 Intertemporal Resource Constraints
Households are subject to the resource constraints, meaning that they cannot alwaysconsume as much as they want. Suppose that one household is established whent = 0 with population L(0) and initial assets A(0) 1 . The population grows at a rate ofn, and at time T >> 0 the household is dissolved. Nobody dies for t ∈ [0,T].
There are following resources the household is able to manage during its life span:
• Everybody begins to work immediately after birth and never stops working beforeT, receiving a wage w(t) for each moment t ∈ (0,T);
• The household has a flow of consumption C(t) for t ∈ (0,T);• The household rents its assets in an exogenous assets market, and at each moment
t ∈ (0,T) it gets a return r(t)A(t) given the market rate r(t) and the level of the assetsholding A(t) at this moment 2 ;
• And one has to keep in mind that since the population of the household growsat a rate n, per capita assets are actually shrinking at the same rate because thehousehold’s assets have to be equally distributed among all its members. We willsee this effect later.
One can think about the household’s resource constraint in two ways:
• From its life-time point of view what it consumes in its life cannot exceed what itearns. Therefore we can express its life-time budget constraint as
1 Conventionally people use capital letter for aggregate value, and small letter for per capitavalue. What’s more, people use small letter with hat, e.g. k, to denote the value for per effectivelabor. We will see this in S 3.2 This assumption already integrated the possibility that the household may borrow via debtcontract – debts can simply be treated as negative assets with the same market rate r(t).
4
T∫
0
exp
−t∫
0
r(s)ds
C(t)dt ≤ A(0) +
T∫
0
exp
−t∫
0
r(s)ds
w(t)L(t)dt. (1)
Note that the inequality above is expressed as present value at t = 0;• From an instantaneous point of view, at any moment t ∈ (0,T) what it earns less
what it loses becomes the increment in its assets holding. Therefore we can expressits flow budget constraint (also called law of motion) as
A(t) ≤ w(t)L(t) + r(t)A(t) − C(t) (2)
with the boundary conditions
A(t = 0) = A(0),
exp
−t∫
0
r(s)ds
A(t = T)≥ 0.
The latter says that the household is not allowed to end up with strictly positive debt.
It’s fairly trivial to write down such budget constraints in the household level. How-ever, to stand in line with our representative agent argument we have to adaptthese constraints into the individual level, i.e. in per capita terms, and this is lesstrivial to see. Now from (1) using the fact that the population grows exponentially,L(t) = L(0)ent, we can express the household’s life-time budget constraint in per capitaterms
T∫
0
exp
−t∫
0
r(s)ds
c(t)L(0)entdt≤ a(0)L(0) +
T∫
0
exp
−t∫
0
r(s)ds
w(t)L(0)entdt,
T∫
0
exp
−t∫
0
[r(s) − n]ds
c(t)dt≤ a(0) +
T∫
0
exp
−t∫
0
[r(s) − n]ds
w(t)dt.
From (2) we can also express the household’s flow budget constraint in per capitaterms. Note that using log-linearization
For the budget constraints in either household level or individual level, the two typesof expressions, i.e. life-time constraints and flow constraint, seem to be quite differentfrom each other in forms, although the reasoning behind both sounds almost equallyundoubtable. But do they really contain the same information, or, are they reallyinterchangeable when they appear in optimization problems as budget constraints?Well, if T < +∞, i.e. if the end point T is finite time, these two expressions areequivalent. But people do have to worry when T goes to infinity (in this case there isno end-point boundary condition). We will see the deep reason behind this issue inthe later lectures, and readers can already find the lengthy mathematical argumentin one example of A A.2.4.
Writing down the right intertemporal resource constraints is very important in solv-ing dynamic problems. We will see many similar constraints later concerning differentkinds of agents (consumers, firms, governments. . . ) in different settings (discrete orcontinuous time, models with money, debt, bonds, international trade, etc.), and onereally has to look carefully into the details of the timing structures in order to makeeverything correct. In addition one has to be careful when adapting the aggregateresource constraints into the individual ones (per capita forms).
2.2 Firms
Now let’s have a look of the firms’ problem. Shortly speaking a firm in an economyarranges its production with a certain technology in order to maximize its profit. Thefollowing sections discuss how to model these feasibilities and motivations.
6
2.2.1 Technology
The firms adopt the technology described by the neoclassical production function F withcapital K 3 and labor L as input, i.e. output Y = F(K,L) : R2 → R. The productionfunction is neoclassical in the sense that it fulfills the following (relatively mild)assumptions:
(1) Constant return to scale (CRS) If we replicate a factory by doubling the capitaland labor input, the output is also doubled:
F(λK, λL) = λF(K,L),∀λ ∈ R++.
Therefore F(K,L) is homogenous of degree one in K and L.(2) Diminishing marginal return ∀K,L ∈ R++ F(K,L) exhibits
∂F∂K
> 0,∂2F∂K2 < 0,
∂F∂L
> 0,∂2F∂L2 < 0.
(3) Inada conditions We’ll see the importance of the conditions later.
limK→0
∂F∂K
= limL→0
∂F∂L
= +∞,
limK→+∞
∂F∂K
= limL→+∞
∂F∂L
= 0.
These three assumptions directly lead to the following property of neoclassical pro-duction functions:
Proposition 2.1 (Essentiality) Each input is essential in the sense that
F(0,L) = F(K, 0) = 0.
Proof As a first step, it’s easy to see that
limK→+∞
FK
= limK→+∞
∂F∂K
1= 0
by (1) L’Hopital rule (Why is L’Hopital rule plausible here, i.e. why limK→+∞ F(K,L) =+∞?) and (2) Inada condition. Then
3 Please keep in mind that K and L are functions of time t, i.e. K(t) and L(t). We drop t whereit doesn’t lead to confusions.
7
limK→+∞
FK
= limK→+∞
F(1,
LK
)= F(1, 0) = 0
by constant return to scale. And
F(K, 0) = KF(1, 0) = 0
by applying again the assumption of constant return to scale. Similar argument holdsfor proving F(0,L) = 0. 2
2.2.2 Profit Maximization
As we learned in intermediate micro the firms maximize their profit by
maxK,L
Π = F(K,L) − rK − δK − wL
in which r is the real interest rate, δ is the depreciation rate of the capital, and w isthe wage rate for employees. Therefore, R = r + δ defines the rental rate for the firmsto get the capital. Suppose that firms obtain capital and labor from correspondingcompetitive markets in which r and w are determined as equilibrium prices.
The first order conditions require that 4
∂Π
∂K=∂F∂K− r − δ = 0,
∂Π
∂L=∂F∂L− w = 0
meaning that the firms set the input level of capital exactly at which the marginalproduct of capital is equal to the marginal cost, and the input level of labor is exactlyset at the point where the marginal product of labor is equal to the marginal cost.
Again as we did in the section for households the problems concerning firms canalso be expressed in per capita terms. For example per capita capital intensity k = K
L ,as well as per capita output
y =YL
=F(K,L)
L= F
(KL, 1
)= F(k, 1)
4 For a multi-variate function F (x1, . . . , xn) : Rn → R, as a convention in economics, peopleoften denote its partial derivative with respect to the k-th (1 ≤ k ≤ n) variable by Fxk :=∂F(x1,...,xn)
∂xk, or Fk := ∂F(x1,...,xn)
∂xk. For example, people write ∂F
∂K as FK or F1 (and ∂2F∂K∂L as FKL or
F12). In these class notes the author prefers to write all partial derivatives in explicit forms.However, readers are asked to get used to this convention while reading the other literature.
8
from the fact that F(K,L) is homogenous of degree 1. And F(k, 1) simply means theoutput generated by per capita capital input and per capita labor input, which isdefined as the production function in per capita form
y = f (k) = F(k, 1).
As an exercise readers can verify that f (k) has the neoclassical properties such that
• Constant return to scale (CRS)
f (λk) = λ f (k),∀λ ∈ R++.
Therefore f (k) is homogenous of degree one in k.• Diminishing marginal return ∀k ∈ R++, f (k) exhibits
f ′(k) > 0, f ′′(k) < 0.
• Inada conditions If F(K,L) fulfills Inada condition, then f (k) fulfills as well.
limk→0
f ′(k) = +∞,lim
k→+∞f ′(k) = 0.
Rewrite the first order conditions in terms of per capita variables
r =∂F∂K− δ
=∂FL∂k− δ
= f ′(k) − δ,
and by the Euler’s formula
F(K,L) =∂F∂K
K +∂F∂L
L,
f (k) = k f ′(k) − w,
one can see that
w = f (k) − k f ′(k).
9
2.3 Technological Progress
Technological progress is a huge dimension in considering sustainable economicgrowth. The simplest way (as we do for most of the time in this course) to integratetechnological progress into a macro model is to modify the production function –basically one can easily imagine the following three possible treatments:
(1) We assume that technological progress helps to magnify the output in the formof Hicks-neutral production function
Y(t) = A(t)F (K(t),L(t))
in which A(t) ∈ R++,∀t is the function capturing technological progress;(2) We assume that technological progress helps to save the input of capital in the
form of Solow-neutral (or capital-augmenting) production function
Y(t) = F (A(t)K(t),L(t)) ;
(3) We assume that technological progress helps to save the input of labor in theform of Harrod-neutral (or labor-augmenting) production function
Y(t) = F (K(t),A(t)L(t)) .
However people mostly use the last form (labor-augmenting production function) inpractices. The deep reason is that for most of the settings this is the only form thatensures the existence of a steady state in dynamic analysis. See Uzawa (1961) andSchlicht (2006).
3 A Simple Dynamic Partial Equilibrium Model (Solow-Swan Model)
Solow (1957) and Swan (1956) consider an economy with exogenous saving rates ∈ [0, 1], and all the other parameters are endogenously determined – that’s why itis a partial equilibrium model. Suppose that at each moment t, Y(t) = F (K(t),A(t)L(t))output is produced via a neoclassical, labor augmenting production function. A share1 − s of Y(t) is consumed as C(t) = (1 − s)Y(t), and s of Y(t) is saved as investment.Moreover,
• The depreciation rate for capital stock K(t) is δ;• Technological progress index A(t) grows at a rate g;• Population grows at a rate n.
In the end what is left in this economy becomes the change in capital stock:
10
K(t) = I(t) − δK(t),= F (K(t),A(t)L(t)) − C(t) − δK(t),= sF (K(t),A(t)L(t)) − δK(t).
From k(t) = K(t)A(t)L(t) by log-linearization
˙k(t)
k(t)=
K(t)K(t)− A(t)
A(t)− L(t)
L(t),
˙k(t) =K(t)K(t)
K(t)A(t)L(t)
− (n + g)k(t)
=K(t)
A(t)L(t)− (n + g)k(t),
then insert the expression for K(t)
˙k(t) =sF (K(t),A(t)L(t)) − δK(t)
A(t)L(t)− (n + g)k(t),
= s f(k(t)
)− (δ + n + g)k(t). (3)
k*
k
ˆf k
ˆn g k
c
Gross investment
Net investment
0k
0k
ˆsf k
Fig. 1. T D k
11
Next, as Romer (2006), one can analyze the economic dynamics in a graphical ap-proach, as F 1 shows, with the steady state value k∗ as the economy’s long-runequilibrium. However, merely with function s f
(k(t)
)being concave, there may be
cases other than F 1. For example, in F 2 (a) the curve s f(k(t)
)lies below
(δ + n + g
)k(t), and in F 2 (b) the curve s f
(k(t)
)converges to a line paralell to(
δ + n + g)
k(t) — in these two cases, there exists no steady state k∗ > 0.
kgn ˆ
kgn ˆ
ksf ˆ
ksf ˆ
0 0k k
)(a )(b
Fig. 2. O C C
The dynamic system is made deterministic by adding Inada conditions, which en-sures a unique steady state k∗ > 0. To show this, rewrite equation (3) as
˙k(t)
k(t)=
s f(k(t)
)
k(t)− (δ + n + g).
Then we claim that
(1) Functions f(k(t))
k(t)is monotonically decreasing with k(t) (Why?);
(2) By Inada conditions, limk(t)→0s f(k(t))
k(t)= +∞ and limk(t)→+∞
s f(k(t))k(t)
= 0 (Why?);
(3) By claims 1 and 2, there exists a unique k∗ > 0 such that˙k(t)k(t)
= 0 as F 3 shows(Why?).
Further Issues (To be discussed in the class):
• Dynamics and path of convergence (F 1)• Golden rule and dynamic (in-)efficiency (F 4)• Stability• Speed of convergence• Comparative statics
12
gn
k
ksf
ˆ
ˆ
0 k*
k
Fig. 3. C I C
(n � �) �
Initial increaseof
Dynamically inefficient region
n � �
s2 � f ( )
sgolden � f ( )
golden
*1
*2
*2
golden
s1 � f ( )
f ( )
n g
slope n g
k
k
k
k
k
k kk k
c
c
c
Fig. 4. T G R (Stolen from Barro and Sala-ı-Martin (2004), p. 36)
• Continuous- versus Discrete-time approach
13
4 Readings
Romer (2006) Chapter 1, or Barro and Sala-ı-Martin (2004) Chapter 1.
5 Bibliographic Notes
Most of the material presented in this lecture can be found in the introductory chaptersof every textbook on advanced macroeconomics or economic growth, e.g. Acemoglu(2009), Barro and Sala-ı-Martin (2004), Romer (2006), Solow (2000), just name a few.Further discussions on microeconomic theory can be found in the classics like Mas-Colell et al. (1995), Varian (1992), or the soon-to-be classic Rubinstein (2006).
Prerequiste of analytical techniques in this course is first-year undergraduate math-ematics, roughly equivalent to Strang (1991) which includes essentials of calculus,basic knowledge of linear algebra as well as probability and statistics (as Hogg et al.(2004)). In the appendix of each chapter readers find fundamental mathematical factswhich one may need to read through the text.
There are many handbooks from which reader may get some quick references. Zeidleret al. (2003a, 2003b, 2004) work well for most general questions, however, economistsmay prefer the special design of Sydsæter, Strøm and Berck (2005). For one who wantsto go into technical details, there is a pretty wide collection of excellent textbooks foreconomic analysis, such as (at an increasing level of difficulty and analytical rigor)Hoy et al. (2001), Sydsæter, and Hammond (2005), Chiang et al. (2005), Simon et al.(1994), de la Fuente (2000) and Sydsæte, Hammond, Seierstad and Strøm (2005).Above all, Kolmogorov and Fomin (1970), Rudin (1976) provide readers relativelyeasier entrance to advanced mathematical analysis.
6 Exercises
6.1 Solow-Swan Model
As in the standard Solow-Swan model, assume that both labor and capital are paidtheir marginal products and the production follows labor-augmenting neoclassicalproduction function. Let w denote ∂F(K,AL)
∂L and r denote ∂F(K,AL)∂K − δ.
a) 5 Show that the marginal product of labor is w = A[ f (k) − k f ′(k)].
5 The level of difficulty. A is the lowest.
14
b) Show that if both capital and labor are paid their marginal products, constantreturn to scale imply that the total amount paid to the factors of production equalstotal net output, i.e. wL + rK = F(K,AL) − δK.
c) What are the growth rates of r and w on a balanced growth path? Show that thismodel exhibits the properties as Kaldor facts, such that r is roughly constant over time,as are the shares of output going to capital and to labor.
d) Suppose that the economy starts with a level of k less than k∗. As the time goingon, is w growing at a rate greater than, less than, or equal to its growth rate on thebalanced growth path? What about r?
6.2 Solow-Swan Model with Human Capital
(Hall & Jones, 1999) Suppose, similar as standard Solow-Swan model, in an economywith constant population growth rate L
L = n as well as constant exogenous techno-logical progress rate A
A = g, the output Y is produced with physical capital K andhuman capital H, according to a Cobb-Douglas technology
Y = Kα(AH)1−α, 0 < α < 1.
Human capital is accumulated by workers (raw labor, L) by investing into educationor training. Suppose that individuals spend a constant fraction of their time, u,learning and that skills are accumulated according to the following expression
H = eψuL,
where ψ measures the percentage increase in H following a small increase in u, i.e.d ln H
du = ψ. (Note that if u = 0, H = L and all labour is unskilled). The rest of theeconomy is as in the standard Solow-Swan model. Physical capital obeys the law ofmotion
K = sY − δK.
a) Express the production function in terms of output per effective worker.
b) What is the fundamental equation of motion for this economy? Illustrate thedynamic behaviour of the system using a diagram.
c) Find the steady state values of output per effective worker and output per capita.
15
d) Suppose a country decides to increase the fraction of time devoted to education,u. What will happen to output per capita in steady state, ∂y∗
∂u ?
e) Show the transition to the new steady state in the phase diagram.
6.3 Solow-Swan Model with Minimum Wage
Consider a Solow-Swan economy with firms adopting Cobb-Douglas technology
F(K(t),L(t)) = K(t)α(A(t)L(t))1−α
in which the index of technological progress A(t) grows at a rate g, starting from A(0).Suppose that the capital depreciates at a rate δ and the labor force grows at a rate n.The saving rate is constant s for all the time.
a) Write down the capital flow in terms of per capita variables.
b) The economy starts with per effective labor capital k(0). Calculate k(t) and showthat
limt→+∞
k(t) = k∗
in which k∗ is the steady state level of capital stock per effective labor.
c) What fraction of growth in YL does the growth accounting framework attribute to
growth in KL ? What fraction to technological progress?
d) How can you reconcile this finding with the fact that the Solow model impliesthat the growth rate of Y
L on the balanced growth path is solely determined by theexogenous rate of technological progress?
e) If we increase the saving rate s by ten percent, how will the steady state outputper effective labor, f (k∗), change?
f) Show that under constant saving rate s the steady-state per capita real wage andconsumption grows at rate g. Now suppose that an economy is already in the steadystate in t = T. The Labor Party proposes the introduction of Minimum Wage Actconcerning a per capita wage increase at T, w > wT, and from T onwards it growsexponentially at rate g (wT is the steady-state value of wage rate at T). Characterizethe evolution of employment, capital, and output for all t > T under the followingtwo different proposals:
(1) The Act is effective forever;
16
(2) The Act is effective till t = T′ > T. And then the minimum wage is adjusted toa new growth rate, i.e. the new minimum wage is defined as w′(t) which growsfrom wT′ at a rate 0 < g′ < g (wT′ is the previous period stipulated minimumwage level at T′). Show that the minimum wage growing at rate g′ initiallyslows down the rise in unemployment and later on leads to increasing levelsof employment until full employment is reestablished. Argue that at this datethe minimum wage ceases to be binding and that the actual wage per effectivelabor as well as the capital stock per effective labor is lower than their initiallaissez-faire level.
6.4 Solow-Swan Model with Endogenous Labor Force Participation
Consider the standard Solow-Swan model in which the population Lt grows at aconstant exponential rate, i.e. Lt
Lt= nL. Abstract from technological progress and
assume that the labor force participation rate is a function of the real wage rate wt,according to
p (wt) =Nt
Lt,
where Nt is employment. Assume that the production function is Cobb-Douglas,
Yt = Kαt N1−α
t , 0 < α < 1.
a) Develope the fundamental differential equation for the per capita capital stockkt = Kt
Ltand show tha it depends on the elasticity of the participation rate with respect
to the wage ηpw and on the elasticity of wages with respect to per capita capital ηwk.
b) What are the likely signs of ηpw and ηwk? Explain intuitively.
c) Explain both formally and intuitively what the effect of an endogenous partici-pation rate is on the adjustment speed of the economy.
6.5 Solow-Swan Model with Endogenous Heterogeneity in Technology Diffusion
Instead of Solow-Swan model in an individual economy, suppose that the worldeconomy consists of J countries, indexed j = 1, ..., J , each with access to a neoclassicalaggregate production function for producing a unique final good,
17
Y j(t) = F(K j(t),A j(t)L j(t)
),
where Y j(t) is the output of this unique final good in country j at time t, and K j(t)and L j(t) are the capital stock and labor supply. Finally, A j(t) is the technology of thiseconomy, which is both country-specific and time-varying. To ease our discussionsin the following, define per capita income as well as the effective capital-labor ratioin country j at time t as
y j(t) =Y j(t)L j(t)
,
k j(t) =K j(t)
A j(t)L j(t).
Suppose that time is continuous, that there is population growth at the constant raten j ≥ 0 in country j, and that there is an exogenous saving rate equal to s j ∈ (0, 1)in country j and a depreciation rate of δ ≥ 0 for capital. Define the growth rate oftechnology of country j at time t as
g j(t) =A j(t)A j(t)
,
and the initial conditions are k j(0) > 0 and A j(0) > 0 for each j = 1, ..., J.
a) Derive the law of motion of k j(t) for each country.
Now let us assume that the worlds technology frontier, denoted by A(t) = max{A1(t), ...,AJ(t)
},
grows exogenously at the constant rate
g(t) =A(t)A(t)
> 0
with an initial condition A(0) > 0. Moreover, each countrys technology progresses asa result of absorbing the worlds technological knowledge. In particular, let us positthe following law of motion for each countrys technology:
A j(t) = σ j
(A(t) − A j(t)
)+ λ jA j(t)
where σ j ∈ (0,+∞) and λ j ∈ [0, g) for each j = 1, ..., J.
b) Provide some intuitions for the law of motion above.
c) Define the measure of country j’s distance to the world technology frontier as
18
a j(t) =A j(t)A(t)
.
Show that
a j(t) = σ j −(σ j + g − λ j
)a j(t). (4)
d) The world’s equilibrium is the sequence{[
k j(t), a j(t)]+∞
t=0
}J
j=1such that the law of
motion in a) as well as equation (4) are both satisfied. A steady-state world equilib-rium is then defined as a steady state of this equilibrium path, that is, an equilibriumwith k j(t) = a j(t) = 0 for each j = 1, ..., J. Show that there exists a unique, globallystable steady-state world equilibrium in which income per capita in all countriesgrows at the same rate g > 0. Moreover, for each j = 1, ..., J, compute the steady-state
world equilibrium{k∗j, a
∗j
}J
j=1. Does your result imply that all countries will converge
to the same level of income per capita?
References
A, D. (2009): Introduction to Modern Economic Growth. Princeton: PrincetonUniversity Press (forthcoming).
B, R. J. X. S-ı-M (2004): Economic Growth (2nd Ed.). Cambridge:MIT Press.
C, A. C. K. W (2005): Fundamental Methods of Mathematical Eco-nomics (4th Ed.). Boston: McGraw-Hill Irwin.
F, A. (2000): Mathematical Methods and Models for Economists. New York:Cambridge University Press.
H, R. E. C. I. J (1999): “Why Do Some Countries Produce So Much MoreOutput Than Others?”Quarterly Journal of Economics, 114, February, 83–116.
H, R. V., MK, J. W. A. T. C (2004): Introduction to Mathematical Statis-tics (6th Ed.). New Jersey: Prentice-Hall.
H, M., L, J., MK C., S, T. R. R (2001): Mathematics forEconomics (2nd Ed.). Cambridge: MIT Press.
K, A. N. S. V. F (1970): Introductory Real Analysis. New Jersey:Prentice-Hall.
M-C, A., W, M. D. J. R. G (1995): Microeconomic Theory. NewYork: Oxford University Press.
R, D. (2006): Advanced Macroeconomics (3rd Ed.). Boston: McGraw-Hill Irwin.R, A. (2006): Lecture Notes in Microeconomic Theory: The Economic Agent.
Princeton: Princeton University Press.
19
R, W. (1976): Principles of Mathematical Analysis (3rd Ed.). New York: McGraw-Hill.
S, E. (2006): “A Variant of Uzawa’s Theorem.”Economics Bulletin 5, Decem-ber, 1–5.
S, C. P. L. B (1994): Mathematics for Economists. New York: W. W. Nor-ton & Company.
S, R. M. (1957): “Technical Change and the Aggregate Production Function.”Reviewof Economics and Statistics 39, August, 312–320.
S, R. M. (2000): Growth Theory: An Exposition (2nd Ed.). New York: Oxford Uni-versity Press.
S, G. (1991): Calculus. Wellesley, MA: Wellesley-Cambridge Press.S, T. W. (1956): “Economic Growth and Capital Accumulation.”Economic Record
32, November, 334–361.S, K. P. H (2005): Essential Mathematics for Economic Analysis
(2nd Ed.). New Jersey: Prentice Hall International.S, K., H, P., S, A. A. S (2005): Further Mathemat-
ics for Economic Analysis. New Jersey: Prentice Hall International.S, S, A. P. B (2005): Economists’ Mathematical Manual (4th Ed.).
Heidelberg: Springer Verlag.V, H. R. (1992): Microeconomic Analysis (3rd Ed.). New York: W. W. Norton &
Company.U, H. (1961): “Neutral Inventions and the Stability of Growth Equilibrium.”Review
of Economic Studies 28, April, 117–124.Z, E. , G, G. I. N. B (H.) (2003): Teubner – Taschen-
buch der Mathematik 1 (2. Aufl.) . Stuttgart: Teubner Verlag.Z, E. , G, G. I. N. B (H.) (2003): Teubner – Taschen-
buch der Mathematik 2 (8. Aufl.) . Stuttgart: Teubner Verlag.Z, E. , G, G. I. N. B (E.) (2004): Oxford Users’ Guide to
Mathematics. New York: Oxford University Press.
Appendix
A Useful Results of Mathematics
A.1 Homogenous Function
For any scalar r the real-valued function F(x1, x2, . . . , xn) : Rn → R is homogeneous ofdegree r if
Since λ is arbitrarily taken, the equation above surely holds when λ = 1. 2
21
A.2 First-Order Ordinary Differential Equations
A.2.1 Homogenous Equation
Equation of the following form is called homogenous equation
x + Ax = 0
where A is a constant, since the only constant term is 0 on the right hand side.
To solve it, rearrange the equation as
xx
=−A,
d ln xdt
=−A,
integrate with respect to t and get the general solution
x(t) = exp(−At + c).
Suppose it’s known that x(0) = x0, then substitute for c and get the special solution
x(t) = x0 exp(−At).
A.2.2 Linear Differential Equation with Propagator
Equation of the following form is called linear differential equation with propagator
x = A(t)x + B(t) (A.1)
where both A and B are functions of time and B(t) is an additional term without x.The equation is called autonomous when B(t) is a constant, i.e. its dependence on timeonly shows up throught the terms concerning x(t).
The solution method, variation of constants, was proposed by Lagrange (1736-1813).Start from solving homogenous problem
x = A(t)x,∫d ln x =
∫A(t)dt,
22
that is
x(t) = C exp(∫
A(t)dt). (A.2)
Then Lagrange’s idea is that by introducing perturbation term B(t) (called propagatorin physics and engineering) the constant term C in (A.2) becomes time dependent,i.e. C = C(t). Differentiating (A.2) with respect to t gives
x = C exp(∫
A(t)dt)
+ A(t)x. (A.3)
Compare (A.3) with (A.1) and get
C = B(t) exp(−
∫A(t)dt
).
Now it’s simple to solve for C
C =
∫B(τ) exp
(−
∫A(τ)dτ
)dτ + c. (A.4)
Equation (A.1)’s solution is characterized by (A.2) and (A.4). Constant c can be solvedwhen x(0) is known.
A.2.3 Bernoulli Equation
Equation of the following form is called Bernoulli equation
x = A(t)x + B(t)xα.
To solve it, define
y = x1−α
and subsitute for x. Then one gets linear differential equation with propagator
y = (1 − α)A(t)y + (1 − α)B(t).
23
A.2.4 Example
Suppose that a representative infinitely living agent from a infinitely living householdfacing the following problem 6 :
• She has an initial level of assets stock a(0) when she is born at t = 0;• She receives a wage income flow w(t) for t ≥ 0;• She receives a income flow r(t)a(t) from renting her assets for t ≥ 0;• She generates a consumption flow c(t) for t ≥ 0;• The population of the household grows at a rate n, implying that her assets are
vaporizing at the same rate.
Then it’s quite straight-forward to see her life-time budget constraint
a(0) ≥ −+∞∫
0
exp
−t∫
0
[(r(τ) − n)] dτ
[w(t) − c(t)] dt, (A.5)
or perhaps it’s more straight-forward by putting it in another way
+∞∫
0
exp
−t∫
0
[(r(τ) − n)] dτ
c(t)dt ≤ a(0) +
+∞∫
0
exp
−t∫
0
[(r(τ) − n)] dτ
w(t)dt
meaning that the present value (by discounting everything with the market discountrate r and the demographic discount rate, i.e. population growth, n) of her life-timeconsumption should not exceed the present value of her life-time wealth.
It’s also straight-forward to see her flow budget constraint can be written as
Note that there is no boundary constraint for t = +∞ which is directly imposed onthe flow budget constraint.
A very important question is to ask whether this is equivalent to (A.5) (To makelife easier from now on we take equality for both constraints). Note that the flowbudget constraint has exactly the form of a linear ordinary differential equation
6 Readers will learn better interpretations in the lecture. Please concentrate on two questionshere: (1) how to write down a flow budget constraint (law of motion); (2) how to solve thebudget constraint as an ordinary differential equation.
24
with propagator, which suggests that we may check the equivalence by solving thisdifferential equation.
First solve the equation without the propagator w(t) − c(t) (simply a homogenousequation)
a(t) = [r(t) − n] a(t),d ln a(t)
dt= r(t) − n,
a(t) = C exp
t∫
0
[(r(τ) − n)] dτ
.
Then take account of the effect from the propagator by setting constant C as a functionof t, C(t), and take derivative with respect to t
a(t) = C(t) exp
t∫
0
[(r(τ) − n)] dτ
+ C(t) exp
t∫
0
[(r(τ) − n)] dτ
︸ ︷︷ ︸
a(t)
[(r(t) − n)]
= C(t) exp
t∫
0
[(r(τ) − n)] dτ
+ [(r(t) − n)] a(t).
Compare it with the original equation
a(t) = [r(t) − n] a(t) + w(t) − c(t)
one can see that
C(t) exp
t∫
0
[(r(τ) − n)] dτ
= w(t) − c(t).
Solve for C(t) by integrating both sides with respect to t
C(t) = c +
t∫
0
exp
−s∫
0
[(r(τ) − n)] dτ
[w(s) − c(s)] ds.
Insert it into our interim result
25
a(t) = C(t) exp
t∫
0
[(r(τ) − n)] dτ
and we obtain
a(t) = c exp
t∫
0
[(r(τ) − n)] dτ
+
t∫
0
exp
t∫
s
[(r(τ) − n)] dτ
[w(s) − c(s)] ds.
By applying that a(t = 0) = a(0) solve to determine the constant c = a(0). Therefore
a(t) = a(0) exp
t∫
0
[(r(τ) − n)] dτ
+
t∫
0
exp
t∫
s
[(r(τ) − n)] dτ
[w(s) − c(s)] ds.
In terms of a(0), it can be written as
a(0) = a(t) exp
−t∫
0
[(r(τ) − n)] dτ
−t∫
0
exp
−s∫
0
[(r(τ) − n)] dτ
[w(s) − c(s)] ds.
Alas, it seems different from (A.5)! These two constraints are NOT equivalent!
Later you will know that for the problems like this, as a result of optimization, thetransversality condition leads to
limt→+∞
a(t) exp
−t∫
0
[(r(τ) − n)] dτ
= 0,
implying that
a(0) = −+∞∫
0
exp
−t∫
0
[(r(τ) − n)] dτ
[w(t) − c(t)] dt.
And this is exactly the life-time constraint (A.5). Now we learned our first lessonfrom this exercise: The optimality conditions from the optimal control theory (whichyou will learn in the next class) require the transversality condition, making the twobudget constraints interchangeable. The good news is that in the exercise of solving
26
optimization problems you are allowed to use the flow budget constraint instead ofthe life-time budget constraint.
In the later lectures you will hear the so-called No-Ponzi-Game constraint saying that
limt→+∞
a(t) exp
−t∫
0
[(r(τ) − n)] dτ
≥ 0,
which shall be added as a constraint in the very beginning your optimization exerciseto rule out some economically implausible paths, and the transversality conditionsays that such contraint is in fact binding in the optimum.
From the solution procedure it is pretty clear to see why such constraint has to beimposed. Hopefully this gives readers the necessary strict (unfortunately, less enter-taining) mathematical reasoning beyond the anecdotes of the famous gambler. 2
A.3 Miscellaneous
Implicit Function Theorem Implicit function is defined through equation F(x1, x2, . . . , xn,u) =0, x1, x2, . . . , xn,u ∈ R and u(x1, x2, . . . , xn) : Rn → R. Given
∂F∂u
, 0
then
∂u∂xi
:= uxi = −∂F∂xi
∂F∂u
,∀i ∈ {1, . . . , n}.
L’Hopital’s Rule Suppose that f (x) : R→ R and g(x) : R→ R are twice continuouslydifferentiable in the neighborhood of x∗ where
limx→x∗
f (x) = limx→x∗
g(x) = 0
or
limx→x∗
f (x) = limx→x∗
g(x) = +∞.
Then
limx→x∗
f (x)g(x)
= limx→x∗
f ′(x)g′(x)
.
Leibnitz’s Rule Suppose that function f (x) is defined as
27
f (x) =
u2(x)∫
u1(x)
g(t, x)dt, x ∈ [a, b].
Suppose that• g(t, x) and ∂g
∂x are continuous in both t and x for t ∈ [u1(x),u2(x)] and x ∈ [a, b], aswell as
• u1(x) and u2(x) are continuous and differentiable on x ∈ [a, b],then
ddx
f (x) =
u2(x)∫
u1(x)
∂
∂xg(t, x)dt + g (u2(x), x)
ddx
u2(x) − g (u1(x), x)d
dxu1(x).
As special cases, one can easily see that
ddx
u(x)∫
a
g(t)dt = g (u(x))ddx
u(x),
ddx
a∫
u(x)
g(t)dt =−g (u(x))d
dxu(x),
ddx
x∫
a
g(t)dt = g(x).
Log-Linearization Suppose that functions x1(t), x2(t), . . . , xn(t) are all functions of t.Suppose that function f (t) is the product of xi(t)’s
f (t) = x1(t)x2(t) . . . xn(t),
then if we take logarithm to this equation
ln f (t) = ln x1(t) + ln x2(t) + . . . + ln xn(t)
and take derivative with respect to t, we get
˙f (t)f (t)
=x1(t)x1(t)
+x2(t)x2(t)
+ . . . +xn(t)xn(t)
meaning that the change rate of f (t) is the sum of the rates of xi(t)’s.Similarly, suppose that
g(t) =x1(t)x2(t) . . . xm(t)
xm+1(t)xm+2(t) . . . xn(t)
28
in which 1 ≤ m < n. Then
g(t)g(t)
=x1(t)x1(t)
+x2(t)x2(t)
+ . . . +xm(t)xm(t)
− xm+1(t)xm+1(t)
− xm+2(t)xm+2(t)
− . . . − xn(t)xn(t)
.
And we will see more techniques of log-linearization later.
Taylor Expansion If function f (x) : R→ R is• well defined on closed interval [a, b] and• continuously differentiable till n + 1-th order, i.e. f ′(x), f ′′(x), . . . , f (n+1)(x) exist for
If at (x0, y0) function f (x, y) : R2 → R is continuously differentiable till n + 1-thorder in the neighborhood B(x0, y0), then for (x, y) ∈ B(x0, y0)
f (x, y) =
n∑
k=0
1k!
[(x − x0)
∂∂x
+(y − y0
) ∂∂y
]k
f(x0, y0
)
+1
(n + 1)!
[(x − x0)
∂
∂x+
(y − y0
) ∂∂y
]n+1
f[x0 + θ (x − x0) , y0 + θ
(y − y0
)]
in which θ ∈ (0, 1), and the last term is the residual.Besides these one can get similar equation for function f (x1, x2, . . . , xn) : Rn → R
(n > 2).Example Taylor expansion for f (x) = ex around an arbitrary point x∗ = a.
f (x) = f (a) + f ′(a)(x − a) +12
f ′′(a)(x − a)2 + . . . +1n!
f (n)(a)(x − a)n + . . .
= f (a) + ea(x − a) +12
ea(x − a)2 + . . . +1n!
ea(x − a)n + . . .
– this is the foundation of the log-linearization techniques which we will frequentlyuse later. Especially when x = 1 and a = 0 the equation above collapses to
29
e =10!
+11!
+12!
+ . . . +1n!
+ . . .
– that’s why people call ln := loge “natural logarithm”.Other very useful results can be derived here directly from the first-order ap-
proximation of ex. Take a = 0, then
ex = 1 + x +12
x2 + . . . +1n!
xn + . . .
The first-order approximation gives
limx→0
ex = 1 + x,
limx→0
ln(1 + x) = x.
B Excercises
B.1 Ordinary Differential Equations
Solve the following problems concerning ordinary differential equations.
a) Find the general solution (and the particular solution for those with initial valuesgiven) for each of the following differential equations.
i) x − 2x = 0 with x(0) = 3. (Answer: x(t) = 3e2t.)
ii) x + 4x − 8 = 0 with x(0) = 2. (Answer: x(t) ≡ 2.)
iii) x + 2x − et = 0 with x(0) = 32 . (Answer: x(t) = 1
3et + 76e−2t.)
iv) x = x − x2. (Answer: x(t) = 11−c2e−t .)
v) x + xt = tα. (Answer: x(t) = 1
α+2tα+1 + ct when α , −2, and x(t) = ln t
t + ct when α = −2.)
vi) 7 x + sign(t)x = 0 with t ∈ (−∞,+∞) and x(1) = 1. (Answer: x(t) = e1−|t|.)
7 The sign function is defined as
sign(x) =
−1 for x < 0
0 for x = 0
1 for x > 0
.
30
b) Show that if α > 0 and λ > 0, then for any real β, every solution of
dydx
+ αy(x) = βe−λx
satisfies limx→+∞ y(x) = 0. (The case α = λ requires special treatment.) Find thesolution for β = λ = 1 which satisfies y(0) = 1. Sketch this solution for 0 ≤ x < +∞ forseveral values of α. In particular, show what happens when α→ 0 and α→ +∞.
(Answer: y =βα−λe−λx + ce−αx when α , λ, y = (βx + c)e−αx when α = λ, and easy to see
limx→+∞ y(x) = 0. When β = λ = 1 and y(0) = 1, y = 1α−1e−x + α−2
α−1e−αx. In the limits whenα→ 0
limα→0
y = 2 − e−x,
and when α→ +∞
limα→+∞
y =
1 if x = 0,
0 if x > 0.
Plot y(x) for different αs.)
B.2 Solving Solow-Swan Model
Suppose that the dynamic of the capital intensity k(t) in an economy can be expressedas
k = sAkα − (n + δ)k.
As you can see the economy has a constant saving rate s, and the production followsthe Cobb-Douglas technology y = Akα. What’s more, the capital intensity is erodedby the constant population growth rate n and depreciation rate δ.
Compute the steady-state capital intensity k∗. Suppose that the economy starts fromk(0) < k∗. Compute the time path k(t) and show that limT→+∞ k(T) = k∗. (Hint: This isa Bernoulli equation.)
![rsuper Micro] 7 -e y 7-532-0035 rsuper Micro] Super Micro ...rsuper Micro] 7 -e y 7-532-0035 rsuper Micro] Super Micro E240 cojp sales@deptazett.cojp YoutubeTrazettJP ! r Super Micro]](https://static.documents.pub/doc/80x56/6107e5c6866d1d42da4aa8e4/rsuper-micro-7-e-y-7-532-0035-rsuper-micro-super-micro-rsuper-micro-7-e.jpg)
